## Random walk experiment 1

Practical Activity for 14-16

**Class practical**

Finding displacement after ‘random walk’ using triangular-grid graph paper.

Apparatus and Materials

- Dice, one for each student
- 60-degree isometric graph paper

Health & Safety and Technical Notes

If the pin-through-straw method (teaching note 3) is used, plain steel dressmaking pins would be safer than the larger optical pins

.

Read our standard health & safety guidance

For a good result you will need to take an average of as many trials as possible. It is therefore important that there is enough equipment for all students.

The dice are more likely to be thrown randomly if each one is shaken in a container whose internal dimensions are several times the width of a die.

Procedure

- Each student draws on a sheet of paper six spokes making 60 degrees with the next (the first line being vertical) and labels each direction successively 1, 2, 3, 4, 5, 6.
- The student takes a die, throws it and uses the uppermost number to tell the direction in which to move. Start in the middle of the graph paper and mark a line 1 ‘unit stride’ in that direction.
- Throw the die again and take a further stride in the new direction. Repeat the process until 25 strides have been taken.
- Measure the direct distance from the start to the finish and record this distance on the board.
- When all students have completed steps 1 to 4, preferably several times, calculate a class average of all the results.

Teaching Notes

- A simple analysis of ‘random walk’ shows that the most likely displacement
*R*from the starting point for*N*steps of length _*s*_ is about*s*times √*N*(here 5). That is*R*=*s*√*N*. - With only the relatively small number of trials obtained, even using the whole class several times, the average of all the results is unlikely to give good agreement with that predicted. However, it is very unlikely to be zero displacement and probably a long way from 25 times the stride length for 25 throws. It is more likely to be near enough to the predicted result to lend some support.
- (Since we really should have taken the root-mean-square of the displacement results, the ideal result should be about 0.8 times the arithmetical mean. That is about 4 times
*s*, the stride length.) - Students who have looked at...

...may well grasp that there are millions upon millions of bromine molecules carrying out random walks on a profuse scale. So the statistical prediction will therefore work much better. - Indeed, if students found the average displacement in 500 seconds – the ‘half-brown distance’ – to be about 10 cm, then they are in a position to find a value for the average distance the molecules move between collisions (the mean-free-path). Appealing to measurements of density and pressure of bromine vapour gives the average speed of molecules as 200 metres/second.
- Density of bromine vapour, ρ = 7.5 kg /m
^{ 3 } - Pressure of bromine vapour,
*P*= 10^{ 5 }N/m^{ 2 }(at normal pressure) - Using the relationship
*PV*= 1/3 (*mnv*),^{2 } - Speed of bromine molecules,
*v*= √(3*P*/ ρ) = √(3 x 10^{ 5 }/ 7.5)= 200 m/s - So in 500 seconds the molecules will have travelled a ‘straightened out’ path of 200 x 500 m which is equal to the path between collisions,
*s*, multiplied by the number of collisions,*N*.*200 x 500 = sN*(1) - The random path (the half brown distance) =
*s√N 0.1 = s√N*(2) - From equations (1) and (2) then
*s*= 10^{ -7 }m and*N*= 10^{ 12} - The distance which a molecule travels between collisions is known as its ‘mean free path’.
- Some teachers may then go on to get a rough value for the size of a molecule.
- It is also possible to estimate the time for an air particle to cross a room. See the relevant guidance pages linked below.
- Another version of this investigation, which does not have any restriction on direction, uses a piece of drinking straw 4 cm long, marked at one end, and with a pin through its middle vertically into a small cork. A cross is marked on a sheet of plain paper and the centre of the cork is placed on the cross. The cork is held with one hand and the other hand is used to flick the straw to spin it. When it comes to rest, a mark is made on the paper under the marked end. The cork is moved to the new mark and the procedure repeated, until 25 steps are obtained.

*This experiment was safety-tested in August 2006*